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Abstract Algebra by Robert B Ash by Robert B. Ash

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Abstract Algebra written by Robert B. Ash . This is an other great mathematics book cover the following topics.

  • CHAPTER 0 PREREQUISITES
    Elementary Number Theory, Set Theory, Linear Algebra

  • CHAPTER 1 GROUP FUNDAMENTALS
    Groups and Subgroups, Permutation Groups, Cosets, Normal Subgroups and Homomorphisms, The Isomorphism Theorems, Direct Products

  • CHAPTER 2 RING FUNDAMENTALS
    Basic Definitions and Properties, Ideals, Homomorphisms and Quotient Rings, The Isomorphism Theorems For Rings, Maximal and Prime Ideals, Polynomial Rings, Unique Factorization, Principal Ideal Domains and Euclidean Domains, Rings of Fractions, Irreducible Polynomials

  • CHAPTER 3 FIELD FUNDAMENTALS
    Field Extensions, Splitting Fields, Algebraic Closures, Separability, Normal Extensions,

  • CHAPTER 4 MODULE FUNDAMENTALS
    Modules and Algebras, The Isomorphism Theorems For Modules, Direct Sums and Free Modules, Homomorphisms and Matrices, Smith Normal Form, Fundamental Structure Theorems, Exact Sequences and Diagram Chasing,

  • CHAPTER 5 SOME BASIC TECHNIQUES OF GROUP THEORY
    Groups Acting on Sets, The Orbit-Stabilizer Theorem, Applications to Combinatorics, The Sylow Theorems, Applications of the Sylow Theorems, Composition Series, Solvable and Nilpotent Groups

  • CHAPTER 6 GALOIS THEORY
    Fixed Fields and Galois Groups, The Fundamental Theorem, Computing a Galois Group Directly, Finite Fields, Cyclotomic Fields, The Galois Group of a Cubic, Cyclic and Kummer Extensions, Solvability by Radicals, Transcendental Extensions, Appendix

  • CHAPTER 7 INTRODUCING ALGEBRAIC NUMBER THEORY
    Integral Extensions, Quadratic Extensions of the Rationals, Norms and Traces, The Discriminant, Noetherian and Artinian Modules and Rings, Fractional Ideals, Unique Factorization of Ideals in a Dedekind Domain, Some Arithmetic in Dedekind Domains, p-adic Numbers

  • CHAPTER 8 INTRODUCING ALGEBRAIC GEOMETRY
    Varieties, The Hilbert Basis Theorem, The Nullstellensatz: Preliminaries, The Nullstellensatz: Equivalent Versions and Proof, Localization, Primary Decomposition, Tensor Product of Modules Over a Commutative Ring, General Tensor Products

  • CHAPTER 9 INTRODUCING NONCOMMUTATIVE ALGEBRA
    Semisimple Modules, Two Key Theorems, Simple and Semisimple Rings, Further Properties of Simple Rings, Matrix Rings, and Endomorphisms, The Structure of Semisimple Rings, Maschke’s Theorem, The Jacobson Radical, Theorems of Hopkins-Levitzki and Nakayama

  • CHAPTER 10 INTRODUCING HOMOLOGICAL ALGEBRA
    Categories, Products and Coproducts, Functors, Exact Functors, Projective Modules, Injective Modules, Embedding into an Injective Module, Flat Modules, Direct and Inverse Limits, Appendix

  • SUPPLEMENT
    S1 Chain Complexes, S2 The Snake Lemma, S3 The Long Exact Homology Sequence, S4 Projective and Injective Resolutions, S5 Derived Functors, S6 Some Properties of Ext and Tor, S7 Base Change in the Tensor Product
  • SOLUTIONS TO PROBLEMS

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